Computability in Timed Sets in Opetaa, Estonia Robin Cockett Joaqu - - PowerPoint PPT Presentation

Computability in Timed Sets in Opetaa, Estonia

Robin Cockett Joaqu´ ın D´ ıaz-Bo¨ ıls Jonathan Gallagher Pavel Hrubeˇ s

University ofCalgary

February 4, 2013

Motivation Explicit versus implicit Timed sets Where are we going? Timing maps Complexity orders Restriction categories Computability in timed sets Iteration ... Splitting idempotents Getting non-zero size .. Computability Powerful objects Program objects Turing structure

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Explicit versus implicit

EXPLICIT

(model)

• IMPLICIT

(axiomatic theory) PROBLEM: Need to know what one is modelling ... Need to know how to axiomatize the phenomenon

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Explicit versus implicit

EXPLICIT COMPUTABILITY

(particular model)

• IMPLICIT

COMPUTABILITY

(axiomatic theory)

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Explicit computability

• Turing machine computing (partial) functions
• Kleene’s first model (natural numbers are codes for machines

which act on numbers)

• Oracle computability (jump operators)
• Combinatory and λ-algebras
• Domain theory models.

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Implicit computability

• Axiomatic/logic approaches to computability ...
• Combinatory logic and λ-calculus ...
• Turing categories

Turing categories = abstract computability MANY non-standard models!! ... all models are Turing categories.

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Explicit versus implicit

EXPLICIT COMPLEXITY

(particular model)

• IMPLICIT

COMPLEXITY

(axiomatic theory)

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Explicit complexity

• Time complexity: counting the ticks of a Turing/computing

machine

• Space complexity: counting the storage required by a

Turing/computing machine Want these notions to be independent of the machine model ... Are they? Well not really! e.g. Turing machine versus pointer models at low complexity

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Implicit complexity

Why do it?

• Theoretical understanding of complexity ...

wide variety of different models relationship between different models correspondence between axiomatic features and complexity

• Type checking for complexity

real-time applications ...

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This talk looks at the explicit models complexity theorists themselves use!!! but with categorical eyes! Part of the program of abstract computability: unifies complexity and computability.

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Functional Complexity in a Timed Maps “Universe”

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Functional Complexity in a Timed Maps “Universe”

A surprise connection between partiality and complexity A categorical model/semantics of basic complexity theory A construction that builds models of computability whose total maps are precisely the maps of a given functional complexity class: P-time, Log-space, and above ... I.e. mimic what complexity theorists do ... BUT with categorical eyes.

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The timing of a partial map as a primitive

Start with a notion of timing/costing a partial map: A

f

• |·|f
• B

N f (x) ↓ ⇔ |x|f ↓

• A partial function f may have different timings
• Think of each timing as the cost (time/space/resource) of

computing f by an algorithm

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ASIDE: What is cost?

We shall assume cost is a natural number BUT the theory works more generally! A size monoid is a partially ordered commutative monoid (M, 0, +, ≤) such that

• 0 ≤ x for all x ∈ M,
• x ≤ x′ and y ≤ y′ implies x + y ≤ x′ + y′.

Examples: N,R≥0, N × N... In fact, given any commutative monoid A set x ≤ y if there is a z with x + z = y then x ∼ y ≡ x ≤ y&y ≤ x then size(A) = A/ ∼ is the universal size monoid associated with A. Note: size monoids are orthogonal to commutative groups.

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The Category of Timed Sets

TSet:

• Objects: Sets
• Maps: Timed partial functions
• Identity: The identity function with 0 cost
• Composition:

A

f

• |·|f
• B

g

• |·|g
• C

N N = A

fg

• |·|f +|f (·)|g
• C

N

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The Category of Timed Sets

Too restrictive ... Two maps are equal only if their timing are exactly the same ... Need to capture O( ) the order of complexity ...

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Complexity Orders

An additive complexity order C is a class of monotone functions P : N − → N such that C is:

• down-closed: P ∈ C and Q ≤ P then Q ∈ C;
• closed to composition: if P, Q in C then PQ ∈ C;
• additive: 0 ∈ C and if P, Q ∈ C then P + Q ∈ C.

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Examples of complexity orders

Linear: L = λx.nx|n ∈ N Polynomial: P =

• λx.

n

• i=1

aixi|n ∈ N

• Where denotes down-closure.

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C-ordering

Every complexity order C induces a preorder enrichment on the maps

• f TSet, f ≤C g:
• g(x) ↓ implies f (x) ↓ and g(x) = f (x);
• there is a P ∈ C such that for all x, |x|f ≤ P(|x|g).

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C-equivalence

C-equivalence is the congruence f =C g if: f ≤C g and g ≤C f E.g. f =L g if |x|f ≤ m|x|g and |x|g ≤ n|x|f

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Partiality: Restriction Categories

For each map f : A − → B: a restriction idempotent f : A − → A such that [R.1] f f = f [R.3] f g = f g [R.2] f g = g f [R.4] f h = fh f

• A general framework for partiality [Cockett and Lack 2002]

E.g. Sets and partial functions: f is domain of definition

• P-categories [Robinson and Rosolini 1988]
• Influential paper by Robert Di Paola and Alex Heller on “dominical

categories” (1986) initiates abstract computability.

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Totality in Restriction Category

Recall that a map in a restriction category is total in case f = 1.

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Timed Sets and Restriction Structure

For TSet, the desired restriction is (f , |·|f ) = (f , |·|f ). However, this is not a restriction structure since [R.1] fails: (f , |·|f ) (f , |·|f ) = (f , |·|f + |·|f ) = (f , |·|f )

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The Restriction Category of Timed Sets

TSet may be quotiented by the congruence =C. Proposition. For any complexity order C, TSet/C is a restriction category where (f , |·|f ) = (f , |·|f ).

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Linking Complexity Order and Partiality

Every restriction category is partial order enriched by f ≤ g: f g = f . Lemma. In TSet/C, f ≤ g if and only if g ≤C f .

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Iteration in a restriction category f ⋆ g :

f ng (for at most one n) Intuitively g ⊔ fg ⊔ ffg ⊔ · · ·

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Iteration in a restriction category Iterate(f,g)(x) = while (x in dom(f)) x := f (x) g(x)

Iteration: one way to obtain computability ...

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Disjoint joins

Proposition. For every C, TSet/C has disjoint joins. What does that mean?

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Disjoint joins

Disjointness, means that “domains” do not overlap f g = ∅ The join of disjoint maps f , g is the join, ⊔, with respect to ≤. Must also be ”stable” with respect to composition: h(f ⊔ g) = hf ⊔ hg

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Disjoint joins and iteration

Need disjoint joins for iteration ... f ⋆ g = g ⊔ fg ⊔ ffg ⊔ · · · =

• n

f ng Also need

• n

f ng =C

• n

f ′ng′ whenever f =C f ′ and g =C g′. This requires the complexity order satisfy an extra laxness condition ...

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Distributive Restriction Categories

• Proposition. [Cockett and Lack 2007]

For a restriction category Distributivity ⇒ Extensiveness ⇒ Disjoint Joins

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The Distributive Restriction Category of Timed Sets

Proposition. For every complexity order C, TSet/C is a distributive restriction cate- gory.

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TSet/C has a restriction terminal object

1 = {⋆} is the restriction terminal object. !A : A − → 1 is always defined and has zero cost. Thus, for any f : A − → 1, f = f !A

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Restriction products

The binary restriction product of A, B is A × B with total projections π0, π1 and a unique pairing such that in U

g

• f
• f ,g
• A

A × B

π0

• π1

B

f , g π0 = g f and f , g π1 = f g.

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TSet/C has restriction products

A × B is as in Sets. Projections, π, are always defined and have zero cost. (f , |·|f ), (g, |·|g) := (f , g , |·|f ,g) where |x|f ,g := |x|f + |x|g

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TSet/C has an initial object

0 = ∅ is the initial object. Note also that TSet/C has nowhere defined maps: ∅ := (∅, ∅) : A − → B

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TSet/C has coproducts

A + B is as in Sets. Coprojections σ are always defined and have zero cost. [(f , |·|f ), (g, |·|g)] = ([f , g] , |·|[f ,g]) where |·|[f ,g] = [|·|f , |·|g]

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TSet/C is distributive

The map [A + σB, A + σC] : (A × B) + (A × C) − → A × (B + C) is an isomorphism in Sets, and is zero cost.

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Iteration

f : A − → A g : A − → B f , g disjoint f ⋆ g : A − → B where [Conway 1971]: W.1 (fg) ⋆ h = h ⊔ f ((gf ) ⋆ (gh)) W.2 (f ⊔ g) ⋆ h = (f ⋆ g) ⋆ (f ⋆ h) W.3 (f ⋆ g)h = f ⋆ (gh) W.4 1 × (f ⋆ g) = (1 × f ) ⋆ (1 × g) W.5 f ≤ f ′, g ≤ g′ then f ⋆ g ≤ f ′ ⋆ g′ For example W.1: f ⋆ h = h ⊔ f (f ⋆ h) = h ⊔ f (h ⊔ f (f ⋆ h)) = h ⊔ fh ⊔ f 2(f ⋆ h) = · · ·

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Iteration in TSet/C

Definition. A complexity order C is lax if it is generated by functions P, P(m) + P(n) ≤ P(m + n) Proposition. If C is lax, then TSet/C has iteration Both L and P are lax ..

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Iteration in TSet/C

Given disjoint timed maps f : A − → A, g : A − → B, f ⋆ g(x) :=

• g(f n(x))

∃n.f n ∈ g ↑ else where the cost is |x|f ⋆g :=     

n−1

• i=0

|f i(x)|f + |f n(x)|g ∃n.f n ∈ g ↑ else

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Structural Recap

The basic structural ingredients for building a simple model of complexity:

• Timed functions
• C-equivalence
• Distributivity
• Iteration

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Additional Structure

Discreteness: the map ∆ : A − → A × A has a partial inverse: ∆−1(x, y) =

• x

x = y ↑ else Ranges: restriction idempotents that act on the codomain; provides the image. Finite Joins: If f g = g f , then the stable join with respect to ≤ of f , g exists.

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Total maps

Problems: The total maps are zero cost maps (f , |·|f ) = 1 if, in particular, there is a P such that |·|f ≤ P(0) = 0. However, “running time” should be a function of input size.

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Restriction Idempotents in TSet/C

A restriction idempotent is a timed partial identity Restriction idempotents can be thought of as measuring the size of the input.

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Restriction Idempotents Splitting of TSet/C

An object in Split(TSetC) is a sized set e = (A, |·|e) A map f : e − → e′ is a timed map such that efe′ =C f : |x|e + |x|f + |f (x)|e′ ≤ P(|x|f ) Intuitively a function cannot be “faster” than the time required to read its input and produce its output!!

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Linking Complexity and Totality

Recall, in a restriction category, f is total if f = 1. In the restriction idempotent splitting: f : e − → e′ is total iff e = f In Split(TSetC), what doe this mean? P(|x|e) ≥ |x|f f is C-bounded by the size of its input. i.e. total maps are exactly the “C-timed” maps!!!

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The structure in Split(TSetC)

All the structure lifts to the idempotent splitting. Theorem. Split(TSetC) is a distributive restriction category with iteration where the total maps are precisely those with C-cost.

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Sizes are non-zero ..

Elements with zero size have no impact on complexity! How do we ensure all sizes are non-zero? Answer: Move to the slice Split(TSet/C)/⋆. ⋆ is the subobject 1 = {()} determined by the idempotent ⋆ : 1 − → 1 where |()|⋆ = 1. Lemma If C is a pointed complexity order an object Y ∈ Split(TSet/C) has a total map to ⋆ if and only if each element of Y has a non-zero size.

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Computability

The total maps in Split(TSet/P)/⋆ are by no means the standard PTIME maps of complexity theory:!

• Not computable
• Their P-timing are arbitrarily assigned.

To obtain a standard notion of say PTIME maps we must demand that the maps are realized by a machine. E.g. by a Turing machine with the standard timing. Shall show how this gives a Turing category whose total maps are precisely PTIME maps.

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Powerful and program objects

A is a powerful object in case there are total maps s× : A × A − → A and partial maps P0, P1 : A − → A such that sx P0, P1 = 1A×A. A non-trivial powerful object is List(Bool) with size given by x = 1 + 2 · len(x). There are then linear time maps s×, P0, and P1 which code and decode pairs:

s×(b:bs,b′:bs′)=1:b:1:b′:s×(bs,bs′) P0(1:b: : :rs) = b:P0(rs) s×([], b′ :bs′) = 0:0:1:b′ :s×([],bs′) P0(0 : 0 : : : rs) = [] s×(b : bs, []) = 1 : b : 0 : 0 : s×(bs, []) P1( : :1:b′ : :rs′) = b′ :P1(rs′) P1( : : 0 : 0 : rs′) = []

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Powerful and program objects

Given a powerful object A, an A-program object is an object P which has total operations comp, pair : P × P − → P together with total points Q0, Q1, Q : 1 − → P and a partial evaluation map ev : P × A − → A such that: A

Q,1 P × A ev

• A

A

Q0,1A

• P0
• P × A

ev

A

P × P × A

1×ev

• comp×1 P × A

ev

• P × A

ev

A

A

Q1,1A

• P1
• P × A

ev

A

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Powerful and program objects

P × P × A

π0,π2,π1,π2

• pair×1

P × A

ev

• P × A × P × A

ev×ev

• A × A

s×

A

We shall say that P is a machine program object in case ev = step ⋆ halt where step ∨ halt = 1P×A. In other words ev is a trace of a machine transition which is total.

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Powerful and program objects

A map f : A − → A is said to be P-programmable in case there is an element ⌈f ⌉ : 1 − → P such that P × A ev

A

A

⌈f ⌉,1A

• f
• If X and Y are (particular) retracts of A then a map h : X −

→ Y is P-programmable if the map A

rX

− − → X

h

− − → Y

sY

− − → A is programmable. Theorem If A is an inhabited powerful object in X and P is an A-program

• bject, then the subcategory of P-programmable maps, ProgP(X), on

powers of A forms a cartesian restriction subcategory.

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Turing structure

An object T, in a cartesian restriction category, is a Turing object in case:

• Every object in the category is a retract of T.
• There is an application map, also called a Turing morphism,
• : T × T −

→ T such that for every (partial) map f : T × T − → T there is a total map f : A − → T such that: T × T

• T

T × T

• f ×1T
• f
• A cartesian category with a Turing object is a Turing category:

these provide a unifying formulation of abstract computability. When does an A-program object P make A a Turing object?

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Turing structure

Theorem If X is a cartesian restriction category with an inhabited powerful

• bject A and an A-programming object P such that P is a retract of

A and comp, pair, ev, Q0, Q1, and Q are all P-programmable then ProgP(X) is a Turing category.

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Turing structure

Define the program q := ⌈P0, P1 f ⌉ then A × A

P0,P1×1

A × A × A

1×s×

• rP×s×
• A × A × A

s××1

• 1×s×

A × A

rp×1

P × A

ev

• P × A

sP×1

• ev

A

A × A

s×

• (qsP)×1×1
• A

q×1

• P0,P1

A × A

f

• where (q × 1)sPs× is the required total map and
• := (P0, p1 × 1)(rP × s×)ev.

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Turing Categories and Total Maps

Theorem. The maps that are computable by a Turing machine within P-time in Split(TSetP) form a Turing category, TP, whose Total maps are the P-time maps. Theorem. The maps that are computable on a Transducer within Log-space in Split(TSetL) form a Turing category, TLg, whose Total maps are the Log-space maps.

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Turing Categories and Total Maps

Proof. Turing machines can be composed and paired in P-time in the size of their inputs. For evaluation use the fact that a universal Turing machine can simulate any Turing machine with just a polynomial overhead.

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From Log-space to P-time

There is a restriction preserving functor over Par between the above Turing categories: TLg

• V

TP

• Par

Recall that if T runs in Space(S) then it runs in at most Time(2S).

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And back again?

TLg

• V

TP

?

• Par

V is an isomorphism if and only if P-time = Log-space. If P-time and Log-space are equal, then for all T: Time(T) is Space(Log(T)). Open complexity problem ....

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In Conclusion

Ideas in complexity can now be translated into categorical notions. Open complexity problems have been re-expressed into categorical questions. There are Turing categories whose total maps are precisely those of functional complexity classes. Abstract computability unifies complexity and computability ....

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In Conclusion

For more: Robin Cockett, Joaquin Diaz-Bols, Jonathan Gallagher, Pavel Hrubes “Timed Sets, Functional Complexity, and Computability” Electronic Notes in Theoretical Computer Science Volume 286, 24 September 2012, Pages 117–137.

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